This article deals with the conjugate gradient method on a Riemannian manifold interest in global convergence analysis. The existing algorithms endowed vector transport need assumption that does not increase norm of tangent vectors, order to confirm generated sequences have property. In this article, notion scaled is introduced improve algorithm so may property under relaxed assumption. proposed algorithm, transported rescaled case its has increased during transport. theoretically proved and...

The four-dimensional conformal Kepler problem is reduced by an S1 action, when the associated momentum mapping takes nonzero fixed values. Hamiltonian system proves to be three-dimensional along with a centrifugal potential and Dirac’s monopole field. negative-energy surface turns out diffeomorphic S3×S2, on which symmetry group SO(4) acts. Constants of motion are also obtained, include total angular vector Runge–Lenz-like vector. thus generalized so as admit same group.

Quantum mechanics for internal motions of the three-body system is set up on basis complex vector bundle theory. The called a triatomic molecule in Born–Oppenheimer approximation. states are described as cross sections assigned by an eigenvalue square total angular momentum operator. This equipped with linear connection, which natural consequence geometric interpretation so-called Eckart condition. coupling motion rotation understood naturally terms this connection. Hamiltonian operator...

The problem of the singular value decomposition a matrix can be brought into an optimization on product two Stiefel manifolds different sizes. steepest descent, conjugate gradient, and Newton's methods for are developed applied with several numerical experiments. These algorithms do not need preconditioning that is inevitable in usual algorithm. present method serve to make more accurate obtained by other existing algorithms. In addition, degenerate optimal solutions studied, together...

A several-particle system is called a molecule in the Born–Oppenheimer approximation. The nonrigidity of molecules involves difficulty molecular dynamics. Guichardet [A. Guichardet, Ann. Inst. H. Poincaré 40, 329 (1984)] showed recently that vibration motion cannot general be separated from rotation motion, by using connection theory differential geometry. point his observation center-of-mass made into principal fiber bundle with group as structure group, and equipped Eckart condition...

For a study of the rotational and vibrational motions nonrigid molecular systems in terms differential geometry, quantum dynamics is set up without using Eckart frame. The turn out to induce on internal space ``gauge'' field attached motions. A method obtaining Hamiltonian for molecules presented. wave functions which acts associated with that are simultaneous eigenfunctions P^ J${^}^{2}$, where \ifmmode \hat{J}\else \^{J}\fi{} denote total linear momentum operator angular operator,...

A number of researches have been made for the (Euclidean) Taub-NUT metric, because geodesic this metric describes approximately motion well separated monopole-monopole interaction. From viewpoint dynamical systems, it is known also that admits Kepler-type symmetry, and hence provides a nontrivial generalization Kepler problem. More specifically speaking, U(1) flow system as Hamiltonian reduced to which conserved Runge-Lenz-like vector in addition angular momentum vector, thereby whose...

The author discusses the quantised MIC-Kepler problem and it's symmetry group for negative energies at Lie algebraic level, on basis of complex line bundles.

Quantization of the conformal Kepler problem is defined and studied in order that quantized system, which will be referred to as a hydrogen atom, may associate harmonic oscillator with atom. The atom shares eigenspaces negative energies. four-dimensional reduces three-dimensional ordinary symmetry group SO(4) brought out from subgroup oscillator. gives an example those quantum systems configuration spaces are curved Riemannian nonconstant scalar curvatures Hamiltonian operators depend on curvatures.

According to the Bertrand theorem, Kepler problem and harmonic oscillator are only central force dynamical systems that have closed orbits for all bounded motions. In this article, an infinite number of having such a orbit property found on T*(R3−{0}) by applying slightly modified Bertrand’s method spherical symmetric Hamiltonian with two undetermined functions radius. Actually, any positive rational ν, there exists system just mentioned, which will be called ν-fold system. Each is...

A ’’conformal’’ Kepler problem is defined in order to associate the with harmonic oscillator. The four-dimensional conformal which shares an energy surface oscillator reduces ordinary three-dimensional problem. By use of reduction symmetry group SO(4) brought out from a subgroup SU(2)×SU(2) problem; same as SU(4)

According to the Bertrand theorem, Kepler problem and harmonic oscillator are only central force dynamical systems that have closed orbits for all bounded motions. In this article, other having such a orbit property found on T*(R3−{0}). Consider natural system T*(R4−{0}) whose Hamiltonian function is composed of kinetic potential energies, invariant under SO(2) action. Then one can reduce T*(R3−{0}) by use Kustaanheimo–Stiefel transformation. If original R4−{0} one, Bertrand’s method...

This paper deals with a general method for the reduction of quantum systems symmetry. For Riemannian manifold M admitting compact Lie group G as an isometry group, quotient space Q = M/G is not smooth in but stratified into collection manifolds various dimensions. If action free, made principal fiber bundle structure G. In this case, reduced are set up on associated vector bundles over M/G. idea fails, if free. However, Peter-Weyl theorem works well reducing M. When applied to wave functions...

Reduction by an S1 action is a method of finding periodic solutions in Hamiltonian systems, which known rather as the averaging. Such can be reconstructed orbits pulling back critical points associated ‘‘reduced Hamiltonian’’ on phase space’’ along reduction. For systems two degrees freedom, geometric setting reduction already accomplished case where reduced space two-sphere Euclidean R3, and Hamilton’s equations motion are Euler’s equations. This article deals with will two-hyperboloid...

This paper deals with reduction of the four-dimensional harmonic oscillator by use a one-parameter subgroup U(1) symmetry group SU(4), being generated an ’’angular momentum.’’ The angular momentum determines in energy surface S7 ’’energy-momentum’’ manifold S3×S3 on which SU(2)×SU(2) SU(4) acts. process yields S3×S2 = S3×S3/U(1) SO(4) acts effectively.

In this paper, the problem of finding singular value decomposition (SVD) a complex matrix is formulated as an optimization on product two Stiefel manifolds. A new algorithm for SVD proposed basis Riemannian Newton method. This can provide vectors associated with arbitrary number values from largest one down to smaller one. Furthermore, once sufficiently accurate approximate given, method improve it be computer accuracy permits.